Preheat oven to 375 degrees F (190 degrees C). Lightly grease a 9-inch baking pan.
Unwrap the tube of biscuit dough. Gently separate biscuits; cut each into 4 parts.
Mix sauce and Monterey Jack cheese together in a large bowl. Add biscuit parts; toss to coat. Pour the sauce-biscuit mixture into the prepared baking pan.
Bake in the preheated oven until biscuits are golden, 15 to 20 minutes. Remove pan from oven; top with mozzarella cheese. Continue baking until cheese is melted, about 5 minutes. Cool for 3 minutes before serving.